The ratio of the opposite side to the hypotenuse is called sinus of an acute angle right triangle.
\sin \alpha = \frac(a)(c)
The ratio of the adjacent leg to the hypotenuse is called cosine of an acute angle right triangle.
\cos \alpha = \frac(b)(c)
The ratio of the opposite side to the adjacent side is called tangent of an acute angle right triangle.
tg \alpha = \frac(a)(b)
The ratio of the adjacent side to the opposite side is called cotangent of an acute angle right triangle.
ctg \alpha = \frac(b)(a)
The ordinate of a point on the unit circle to which the angle \alpha corresponds is called sine of an arbitrary angle rotation \alpha .
\sin \alpha=y
The abscissa of a point on the unit circle to which the angle \alpha corresponds is called cosine of an arbitrary angle rotation \alpha .
\cos \alpha=x
The ratio of the sine of an arbitrary rotation angle \alpha to its cosine is called tangent of an arbitrary angle rotation \alpha .
tan \alpha = y_(A)
tg \alpha = \frac(\sin \alpha)(\cos \alpha)
The ratio of the cosine of an arbitrary rotation angle \alpha to its sine is called cotangent of an arbitrary angle rotation \alpha .
ctg\alpha =x_(A)
ctg \alpha = \frac(\cos \alpha)(\sin \alpha)
If \alpha is some angle AOM, where M is a point of the unit circle, then
\sin \alpha=y_(M) , \cos \alpha=x_(M) , tg \alpha=\frac(y_(M))(x_(M)), ctg \alpha=\frac(x_(M))(y_(M)).
For example, if \angle AOM = -\frac(\pi)(4), then: the ordinate of point M is equal to -\frac(\sqrt(2))(2), abscissa is equal to \frac(\sqrt(2))(2) and that's why
\sin \left (-\frac(\pi)(4) \right)=-\frac(\sqrt(2))(2);
\cos \left (\frac(\pi)(4) \right)=\frac(\sqrt(2))(2);
tg;
ctg \left (-\frac(\pi)(4) \right)=-1.
The values of the main frequently occurring angles are given in the table:
| 0^(\circ) (0) | 30^(\circ)\left(\frac(\pi)(6)\right) | 45^(\circ)\left(\frac(\pi)(4)\right) | 60^(\circ)\left(\frac(\pi)(3)\right) | 90^(\circ)\left(\frac(\pi)(2)\right) | 180^(\circ)\left(\pi\right) | 270^(\circ)\left(\frac(3\pi)(2)\right) | 360^(\circ)\left(2\pi\right) | |
| \sin\alpha | 0 | \frac12 | \frac(\sqrt 2)(2) | \frac(\sqrt 3)(2) | 1 | 0 | −1 | 0 |
| \cos\alpha | 1 | \frac(\sqrt 3)(2) | \frac(\sqrt 2)(2) | \frac12 | 0 | −1 | 0 | 1 |
| tg\alpha | 0 | \frac(\sqrt 3)(3) | 1 | \sqrt3 | — | 0 | — | 0 |
| ctg\alpha | — | \sqrt3 | 1 | \frac(\sqrt 3)(3) | 0 | — | 0 | — |
Sinus acute angle α of a right triangle is the ratio opposite leg to hypotenuse.
It is denoted as follows: sin α.
Cosine The acute angle α of a right triangle is the ratio of the adjacent leg to the hypotenuse.
It is designated as follows: cos α.
Tangent acute angle α is the ratio of the opposite side to the adjacent side.
It is designated as follows: tg α.
Cotangent acute angle α is the ratio adjacent leg to the opposite one.
It is designated as follows: ctg α.
The sine, cosine, tangent and cotangent of an angle depend only on the size of the angle.
Rules:
Basic trigonometric identities in a right triangle:
(α – acute angle opposite to the leg b and adjacent to the leg a . Side With – hypotenuse. β – second acute angle).
b | sin 2 α + cos 2 α = 1 | |
a | 1 | |
b | 1 | |
a | 1 1 | |
sin α |
As the acute angle increasessin α andtan α increase, andcos α decreases.
For any acute angle α:
sin (90° – α) = cos α
cos (90° – α) = sin α
Example-explanation:
Let in a right triangle ABC
AB = 6,
BC = 3,
angle A = 30º.
Let's find out the sine of angle A and the cosine of angle B.
Solution .
1) First, we find the value of angle B. Everything is simple here: since in a right triangle the sum of the acute angles is 90º, then angle B = 60º:
B = 90º – 30º = 60º.
2) Let's calculate sin A. We know that the sine is equal to the ratio of the opposite side to the hypotenuse. For angle A, the opposite side is side BC. So:
BC 3 1
sin A = -- = - = -
AB 6 2
3) Now let's calculate cos B. We know that the cosine is equal to the ratio of the adjacent leg to the hypotenuse. For angle B, the adjacent leg is the same side BC. This means that we again need to divide BC by AB - that is, perform the same actions as when calculating the sine of angle A:
BC 3 1
cos B = -- = - = -
AB 6 2
The result is:
sin A = cos B = 1/2.
sin 30º = cos 60º = 1/2.
It follows from this that in a right triangle, the sine of one acute angle is equal to the cosine of another acute angle - and vice versa. This is exactly what our two formulas mean:
sin (90° – α) = cos α
cos (90° – α) = sin α
Let's make sure of this again:
1) Let α = 60º. Substituting the value of α into the sine formula, we get:
sin (90º – 60º) = cos 60º.
sin 30º = cos 60º.
2) Let α = 30º. Substituting the value of α into the cosine formula, we get:
cos (90° – 30º) = sin 30º.
cos 60° = sin 30º.
(For more information about trigonometry, see the Algebra section)
We will begin our study of trigonometry with the right triangle. Let's define what sine and cosine are, as well as tangent and cotangent of an acute angle. This is the basics of trigonometry.
Let us remind you that right angle is an angle equal to 90 degrees. In other words, half a turned angle.
Sharp corner- less than 90 degrees.
Obtuse angle- greater than 90 degrees. In relation to such an angle, “obtuse” is not an insult, but a mathematical term :-)
Let's draw a right triangle. A right angle is usually denoted by . Please note that the side opposite the corner is indicated by the same letter, only small. Thus, the side opposite angle A is designated .
The angle is denoted by the corresponding Greek letter.
Hypotenuse of a right triangle is the side opposite the right angle.
Legs- sides lying opposite acute angles.
The leg lying opposite the angle is called opposite(relative to angle). The other leg, which lies on one of the sides of the angle, is called adjacent.
Sinus The acute angle in a right triangle is the ratio of the opposite side to the hypotenuse:
Cosine acute angle in a right triangle - the ratio of the adjacent leg to the hypotenuse:
Tangent acute angle in a right triangle - the ratio of the opposite side to the adjacent:
Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of the angle to its cosine:
Cotangent acute angle in a right triangle - the ratio of the adjacent side to the opposite (or, which is the same, the ratio of cosine to sine):
Note the basic relationships for sine, cosine, tangent, and cotangent below. They will be useful to us when solving problems.
Let's prove some of them.
Okay, we have given definitions and written down formulas. But why do we still need sine, cosine, tangent and cotangent?
We know that the sum of the angles of any triangle is equal to.
We know the relationship between parties right triangle. This is the Pythagorean theorem: .
It turns out that knowing two angles in a triangle, you can find the third. Knowing the two sides of a right triangle, you can find the third. This means that the angles have their own ratio, and the sides have their own. But what should you do if in a right triangle you know one angle (except the right angle) and one side, but you need to find the other sides?
This is what people in the past encountered when making maps of the area and the starry sky. After all, it is not always possible to directly measure all sides of a triangle.
Sine, cosine and tangent - they are also called trigonometric angle functions- give relationships between parties And corners triangle. Knowing the angle, you can find all its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.
We will also draw a table of the values of sine, cosine, tangent and cotangent for “good” angles from to.
Please note the two red dashes in the table. At appropriate angle values, tangent and cotangent do not exist.
Let's look at several trigonometry problems from the FIPI Task Bank.
1. In a triangle, the angle is , . Find .
The problem is solved in four seconds.
Because the , .
2. In a triangle, the angle is , , . Find .
Let's find it using the Pythagorean theorem.
The problem is solved.
Often in problems there are triangles with angles and or with angles and. Remember the basic ratios for them by heart!
For a triangle with angles and the leg opposite the angle at is equal to half of the hypotenuse.
A triangle with angles and is isosceles. In it, the hypotenuse is times larger than the leg.
We looked at problems solving right triangles - that is, finding unknown sides or angles. But that's not all! IN Unified State Exam options in mathematics there are many problems where the sine, cosine, tangent or cotangent of the external angle of a triangle appears. More on this in the next article.
Sine is one of the basic trigonometric functions, the use of which is not limited to geometry alone. Tables for calculating trigonometric functions, like engineering calculators, are not always at hand, and calculating the sine is sometimes necessary to solve various tasks. In general, calculating the sine will help consolidate drawing skills and knowledge of trigonometric identities.
A simple task: how to find the sine of an angle drawn on paper? To solve, you will need a regular ruler, a triangle (or compass) and a pencil. The simplest way to calculate the sine of an angle is by dividing the far leg of a triangle with a right angle by the long side - the hypotenuse. Thus, you first need to complete the acute angle to the shape of a right triangle by drawing a line perpendicular to one of the rays at an arbitrary distance from the vertex of the angle. We will need to maintain an angle of exactly 90°, for which we need a clerical triangle.
Using a compass is a little more accurate, but will take more time. On one of the rays you need to mark 2 points at a certain distance, set a radius on the compass approximately equal to the distance between the points, and draw semicircles with centers at these points until the intersections of these lines are obtained. By connecting the intersection points of our circles with each other, we get a strict perpendicular to the ray of our angle; all that remains is to extend the line until it intersects with another ray.
In the resulting triangle, you need to use a ruler to measure the side opposite the corner and the long side on one of the rays. The ratio of the first dimension to the second will be the desired value of the sine of the acute angle.
For obtuse angle the task is not much more difficult. We need to draw a ray from the vertex in the opposite direction using a ruler to form a straight line with one of the rays of the angle we are interested in. The resulting acute angle should be treated as described above, sines adjacent corners, forming together a reverse angle of 180°, are equal.
Also, calculating the sine is possible if the values of other trigonometric functions of the angle or at least the lengths of the sides of the triangle are known. Trigonometric identities will help us with this. Let's look at common examples.
How to find the sine with a known cosine of an angle? The first trigonometric identity, based on the Pythagorean theorem, states that the sum of the squares of the sine and cosine of the same angle is equal to one.
How to find the sine with a known tangent of an angle? The tangent is obtained by dividing the far side by the near side or dividing the sine by the cosine. Thus, the sine will be the product of the cosine and the tangent, and the square of the sine will be the square of this product. We replace the squared cosine with the difference between unity and the square sine according to the first trigonometric identity and, through simple manipulations, we reduce the equation to the calculation of the square sine through the tangent; accordingly, to calculate the sine, you will have to extract the root of the result obtained.
How to find the sine with a known cotangent of an angle? The value of the cotangent can be calculated by dividing the length of the leg closest to the angle by the length of the far one, as well as dividing the cosine by the sine, that is, the cotangent is a function inverse to the tangent relative to the number 1. To calculate the sine, you can calculate the tangent using the formula tg α = 1 / ctg α and use the formula in the second option. You can also derive a direct formula by analogy with tangent, which will look like this.
There is a formula for finding the length of the unknown side of any triangle, not just a rectangular one, from two known parties using the trigonometric function of the cosine of the opposite angle. She looks like this.
How to find sine?
Studying geometry helps develop thinking. This subject is necessarily included in school preparation. In everyday life, knowledge of this subject can be useful - for example, when planning an apartment.
The geometry course also includes trigonometry, which studies trigonometric functions. In trigonometry we study sines, cosines, tangents and cotangents of angles.
But for now, let's start with the simplest thing - sine. Let's take a closer look at the very first concept - the sine of an angle in geometry. What is sine and how to find it?
The sine of an angle is the ratio of the values of the opposite side and the hypotenuse of a right triangle. This is a direct trigonometric function, which is written as “sin (x)”, where (x) is the angle of the triangle.
On the graph, the sine of an angle is indicated by a sine wave with its own characteristics. A sine wave looks like a continuous wavy line that lies within certain limits on the coordinate plane. The function is odd, therefore it is symmetrical about 0 on the coordinate plane (it comes out from the origin of the coordinates).
The domain of definition of this function lies in the range from -1 to +1 on the Cartesian coordinate system. The period of the sine angle function is 2 Pi. This means that every 2 Pi the pattern repeats and the sine wave goes through a full cycle.
The values of the sines of the angle are determined using special tables. Such tables have been created to facilitate the counting process complex formulas and equations. It is easy to use and contains meanings not only functions sin(x), but also the values of other functions.
Moreover, a table of standard values of these functions is included in the compulsory memory study, like a multiplication table. This is especially true for classes with a physical and mathematical bias. In the table you can see the values of the main angles used in trigonometry: 0, 15, 30, 45, 60, 75, 90, 120, 135, 150, 180, 270 and 360 degrees.
There is also a table defining the values of trigonometric functions of non-standard angles. Using different tables, you can easily calculate the sine, cosine, tangent and cotangent of some angles.
Equations are made with trigonometric functions. Solving these equations is easy if you know simple trigonometric identities and reductions of functions, for example, such as sin (P/2 + x) = cos (x) and others. A separate table has also been compiled for such reductions.
When the task is to find the sine of an angle, and according to the condition we only have the cosine, tangent, or cotangent of the angle, we can easily calculate what we need using trigonometric identities.
From this equation, we can find both sine and cosine, depending on which value is unknown. We can do it trigonometric equation with one unknown:
From this equation you can find the value of the sine, knowing the value of the cotangent of the angle. To simplify, replace sin 2 x = y and you have a simple equation. For example, the cotangent value is 1, then:
Now we perform the reverse replacement of the player:
Since we took the cotangent value for the standard angle (45 0), the obtained values can be checked in the table.
If you have a tangent value and need to find the sine, another trigonometric identity will help:
It follows that:
In order to find the sine of a non-standard angle, for example, 240 0, you need to use angle reduction formulas. We know that π corresponds to 180 0. Thus, we express our equality using standard angles by expansion.
We need to find the following: sin (180 0 + 60 0). In trigonometry there are reduction formulas that in this case will come in handy. This is the formula:
Thus, the sine of an angle of 240 degrees is equal to:
In our case, x = 60, and P, respectively, 180 degrees. We found the value (-√3/2) from the table of values of functions of standard angles.
In this way, non-standard angles can be expanded, for example: 210 = 180 + 30.
